Natural Number Addition is Cancellable/Proof 1

Theorem

Let $\N$ be the natural numbers.

Let $+$ be addition on $\N$.

Then:

$\forall a, b, c \in \N: a + c = b + c \implies a = b$

$\forall a, b, c \in \N: a + b = a + c \implies b = c$

That is, $+$ is cancellable on $\N$.

Proof

Consider the natural numbers $\N$ defined as a naturally ordered semigroup $\struct {\N, +, \le}$.

By Naturally Ordered Semigroup Axiom $\text {NO} 2$: Cancellability, every element of $\struct {\N, +}$ is cancellable.

$\blacksquare$